Topology

Topology, branch of mathematics that studies properties of geometrical figures or abstract spaces that are independent of shape or distance. Point-set topology deals with ways of defining “nearness” of elements, or points, of a set without necessarily assigning numerical distances to pairs of points. Such a definition is called “a topology on the set,” and the set is called a topological space. The topology makes it possible to define continuous functions on the space. Algebraic topology, or combinatorial topology, uses abstract algebra to treat the ways in which geometrical figures fit together to form figures of higher dimension, disregarding shape. For example a sphere is topologically the same as a cube, but it is distinct from a torus (doughnut) because if the surfaces of the figures are divided into triangles, the algebraic relationships between the triangles will be different in the two cases.